Topic 4 Congruent Triangles

Transcription

1 opic 4 ongruent riangles OI OVVIW VOULY 4-1 ongruent igures nglish/panish Vocabulary udio Online: 4-2 riangle ongruence by and nglish base of an isosceles triangle, p. 168 panish base de un triángulo isósceles base angles of an isosceles triangle, p. 168 ángulos de la base de un triángulo isósceles congruent polygons, p. 148 polígonos congruentes corollary, p. 168 hypotenuse, p. 174 corolario hipotenusa legs of an isosceles triangle, p. 168 legs of a right triangle, p. 174 catetos de un triángulo isósceles catetos de un triángulo rectángulo vertex angle of an isosceles triangle, p. 174 ángulo del vértice de un triángulo isósceles 4-3 riangle ongruence by and 4-4 Using orresponding arts of ongruent riangles 4-5 Isosceles and quilateral riangles 4-6 ongruence in ight riangles 4-7 ongruence in Overlapping riangles IGIL IN and eook ccess Your Homework... ONLIN HOWOK You can do all of your homework online with built-in examples and how e How support! When you log in to your account, you ll see the homework your teacher has assigned you. HOWOK UO YOU IGIL OU OU earsonx X.com o your homework anywhere! You can access the ractice and pplication xercises, as well as Virtual Nerd tutorials, with this Homework utor app, available on any mobile device. UN X N HOWOK HL ccess the ractice and pplication xercises that you are assigned for homework in the tudent ext and Homework Helper, which is also available as an electronic book. 146 opic 4 ongruent riangles

2 heck It Out! 3-ct ath 3-ct ath aybe you ve played this game before: you draw a picture. hen you try to get a classmate to draw the same picture by giving step-by-step directions but without showing your drawing. ry it with a classmate. raw a map of a room in your house or a place in your town. hen give directions to a classmate to draw the map that you drew. ompare your drawings. How similar are they? hink about this as you watch this 3-ct ath video. can page to see a video for this 3-ct ath ask. If You Need Help... VOULY ONLIN You ll find definitions of math terms in both nglish and panish. ll of the terms have audio support. INIV XLOION You ll have access to a robust assortment of interactive explorations, including interactive concept explorations, dynamic activitites, and topiclevel exploration activities. LNING NIION You can also access all of the stepped-out learning animations that you studied in class. UN ONION efer to your notes and solutions in your tudent ompanion. emember that your tudent ompanion is also available as an IVebook accessible on any digital device. INIV H OOL hese interactive math tools give you opportunities to explore in greater depth key concepts to help build understanding. VIUL N Not sure how to do some of the practice exercises? heck out the Virtual Nerd videos for stepped-out, multi-level instructional support. earsonx.com 147

3 4-1 ongruent igures K OU VOULY K (6)() pply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles. K (1)() Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Ě ongruent polygons polygons that have congruent corresponding sides and angles Ě ormulate create with careful effort and purpose. You can formulate a plan or strategy to solve a problem. Ě easonableness the quality of being within the realm of common sense or sound reasoning. he reasonableness of a solution is whether or not the solution makes sense. Ě trategy a plan or method for solving a problem dditional K (1)(), (1)(G) NIL UNNING You can determine whether two figures are congruent by comparing their corresponding sides and angles. Key oncept efinition ongruent igures xample ongruent polygons have congruent corresponding sides and angles. When you name congruent polygons, you must list corresponding vertices in the same order. heorem 4-1 heorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. G H GH G GH G H H hird ngles heorem If... and hen... You will prove heorem 4-1 in xercise Lesson 4-1 ongruent igures

4 roblem 1 How do you know which sides and angles correspond? he congruence statement HIJK LNO tells you which parts correspond. inding ongruent ides and ngles If HIJK LNO, what are the congruent corresponding parts? ides: HI L IJ N JK NO KH OL ngles: H L I J N K O H K J I O N L roblem 2 K rocess tandard (1)() Using ongruent ides and ngles You know two angle measures in. How can they help? In the congruent triangles, corresponds to, so you know that. You can find m by first finding m. ultiple hoice he wings of an -71 lackbird aircraft suggest congruent triangles. What is mj? Use the riangle ngle- um heorem to write an equation involving m. m = 180 olve for m. and are corresponding angles of congruent triangles, so. m = 180 m = 75 m = m = 75 he correct answer is. roblem 3 K rocess tandard (1)(G) inding ongruent riangles How do you determine whether two triangles are congruent? ompare each pair of corresponding parts. If all six pairs are congruent, then the triangles are congruent. re the triangles congruent? Justify your answer. Given = 4 = = 6 =, Given Vertical angles are congruent by the definition of congruent triangles. earsonx.com 149

5 roblem 4 You know four pairs of congruent parts. What else do you need to prove the triangles congruent? You need a third pair of congruent sides and a third pair of congruent angles. roving riangles ongruent Given: L LO, N ON, L O, LN OLN rove: LN LON O tatements easons 1) L LO, N ON 1) Given 2) LN LN 2) eflexive roperty of 3) O, LN OLN 3) Given 4) NL ONL 4) hird ngles heorem 5) LN LON 5) efinition of triangles N ONLIN H O W O K I and LIION XI can page for a Virtual Nerd tutorial video. or additional support when completing your homework, go to earsonx.com. 1. pply athematics (1)() uilders use the king post truss (below left) for the top of a simple structure. In this truss,. List the congruent corresponding sides and angles. G H J 2. he attic frame truss (above right) provides open space in the center for storage. In this truss, G HIJ. List the congruent corresponding sides and angles. L JK. omplete the congruence statements. J 3. L? 4. KJ? 5. K? 6.? 7. L? 8. KJ? OLY@I. List each of the following. 9. four pairs of congruent sides 10. four pairs of congruent angles t an archeological site, the remains of two ancient step pyramids are congruent. find each of the following. (iagrams are not to scale.) GH 13. m GH 14. m m 18. m G 45 ft 128 I L K 45 ft ft G ft H 150 Lesson 4-1 ongruent igures

6 xplain athematical Ideas (1)(G) or xercises 19 and 20, can you conclude that the triangles are congruent? Justify your answers. 19. K and UK 20. Q and UV K U 5 Q 7 6 V U 21. Given: },,, rove: 22. valuate easonableness (1)() andall says he can use the information in the figure to prove. Is he correct? xplain. onnect athematical Ideas ind the measures of the given angles or the lengths of the given sides. 23. m = x + 10, m = 2x 24. m = 3y, m = 6y = 3z + 2, = z = 7a + 5, = 5a If LN, which of the following must be a correct congruence statement?. LN. N L. NL. onnect athematical Ideas (1)() ind the values of the variables. N x 45 4 in. L 2t in. KL K 6x omplete in two different ways: JL?. L J N Z earsonx.com 151

7 31. Given: #, #,,, } rove: 32. nalyze athematical elationships (1)() Write a congruence statement for two triangles. List the congruent sides and angles. 33. Given: }Q, Q, Q, Q bisects rove: Q 34. pply athematics (1)() he 225 cards in racy s sports card collection are rectangles of three different sizes. How could racy quickly sort the cards? 35. onnect athematical Ideas (1)() he vertices of GHJ are G(-2, -1), H(-2, 3), and J(1, 3), and KL GHJ. If L and have coordinates L(3, -3) and (6, -3), how many pairs of coordinates are possible for K? ind one such pair. 36. a. How many quadrilaterals (convex and concave) with different shapes or sizes can you make on a three-by-three geoboard? ketch them. One is shown at the right. b. How many quadrilaterals of each type are there? 37. Given:, rove: Q X est ractice 38. HLN G, m H = 66, and m = 42. What is m? 39. he measure of one angle in a triangle is 80. he other two angles are congruent. What is the measure of each? 40. What is the number of feet in the perimeter of a square with side length 7 ft? 152 Lesson 4-1 ongruent igures

8 4-2 riangle ongruence by and K OU VOULY Ě Number sense the understanding K (6)() rove two triangles are congruent by applying the ide-ngleide, ngle-ide-ngle, ide-ide-ide, ngle-ngle-ide, and Hypotenuse-Leg congruence conditions. of what numbers mean and how they are related K (1)() elect tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. dditional K (1)(), (1)(), (1)(G), (5)(), (5)() NIL UNNING You can prove that two triangles are congruent without having to show that all corresponding sides and angles are congruent. In this lesson, you will prove triangles congruent by using (1) three pairs of corresponding sides and (2) two pairs of corresponding sides and one pair of corresponding angles. ostulate 4-1 ostulate If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. ide-ide-ide () ostulate If...,, ostulate 4-2 ostulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. hen... ide-ngle-ide () ostulate If...,, hen... earsonx.com 153

9 roblem 1 r K rocess tandard (1)() uilding riangles elect a tool to build a triangle with sides of lengths 3 in., 5 in., and 6 in. ompare your triangle with your classmates triangles. What are corresponding sides? orresponding sides are the sides that match and are in the same position. orresponding sides of congruent triangles have the same length. o build a triangle with the given lengths, you might select a real object, such as a straw. ut the straw into pieces of the correct lengths. hread some string through the three pieces of straw, in any order, as shown. ring the ends of the string together and tie them to hold your triangle in place. ake a conjecture about two triangles in which three sides of one triangle are congruent to three sides of the other triangle. oth triangles have the same size and shape. ach triangle fits exactly on top of the other triangle. onjecture: riangles with congruent corresponding sides are congruent. roblem 2 r K rocess tandard (1)() Using You have two pairs of congruent sides. What else do you need? You need a third pair of congruent corresponding sides. Notice that the triangles share a common side, LN. N L Given: L N, L N Gi rove: LN NL r L N LN LN L N Given eflexive rop. of Given LN NL 154 Lesson 4-2 riangle ongruence by and

10 roblem 3 o you need another pair of congruent sides? Look at the diagram. he triangles share. o you already have two pairs of congruent sides. Using What other information do you need to G by? xplain. he diagram shows that G. lso, by the eflexive roperty of ongruence. o prove that G by, you must have congruent included angles. You need to know that G. G roblem 4 Identifying ongruent riangles What should you look for first, sides or angles? tart with sides. If you have three pairs of congruent sides, use. If you have two pairs of congruent sides, look for a pair of congruent included angles. Would you use or to prove the triangles congruent? If there is not enough information to prove the triangles congruent by or, write not enough information. xplain your answer. Use because two pairs of corresponding sides and their included angles are congruent. here is not enough information; two pairs of corresponding sides are congruent, but one of the angles is not the included angle. Use because three pairs of corresponding sides are congruent. Use or because all three pairs of corresponding sides and a pair of included angles (the vertical angles) are congruent. ONLIN H O W O K I and LIION XI can page for a Virtual Nerd tutorial video. or additional support when completing your homework, go to earsonx.com. 1. Use epresentations to ommunicate athematical Ideas (1)() opy the flow chart and complete the proof. Given: JK L, J LK rove: JK LK J K L JK L Given J LK a. K K b. c. d. earsonx.com 155

11 2. elect ools to olve roblems (1)() onsider the following conjecture. If two triangles have the same perimeter, then the triangles are congruent. a. elect a real object that you can use to test the conjecture. xplain your choice. b. Is the conjecture true? If not, make a new conjecture based on your results. xplain your reasoning. 3. xplain athematical Ideas (1)(G) t least how many triangle measurements must you know in order to guarantee that all triangles built with those measurements will be congruent? xplain your reasoning. 4. Given: I GH, H, 5. Given: WZ Z W is the midpoint of GI rove: WZ Z rove: I HG G Z W I H What other information, if any, do you need to prove the two triangles congruent by? xplain. 6. G 7. N L Q U V W 8. valuate easonableness (1)() You and a friend are cutting triangles out of felt for an art project. You want all the triangles to be congruent. Your friend tells you that each triangle should have two 5-in. sides and a 40 angle. If you follow this rule, will all your felt triangles be congruent? xplain. an you prove the triangles congruent? If so, write the congruence statement and name the postulate you would use. If not, write not enough information and tell what other information you would need G Y K J N H V W 156 Lesson 4-2 riangle ongruence by and

12 12. Use epresentations to ommunicate athematical Ideas (1)() ierpinski s triangle is a famous geometric pattern. o draw ierpinski s triangle, start with a single triangle and connect the midpoints of the sides to draw a smaller triangle. If you repeat this pattern over and over, you will form a figure like the one shown. his particular figure started with an isosceles triangle. re the triangles outlined in red congruent? xplain. 13. reate epresentations to ommunicate athematical Ideas (1)() Use a straightedge to draw any triangle JKL. hen construct N JKL using the given postulate. a. b. 14. nalyze athematical elationships (1)() uppose GH JK, HI KL, and I L. Is GHI congruent to JKL? xplain. 15. Given: G}KL, G KL rove: GK KL L K G 16. Given: #, #,, is the midpoint of. rove: X est ractice 17. What additional information do you need to prove that VWY VWZ by?. YW ZW. Y Z. WVY WVZ. VZ VY 18. he measures of two angles of a triangle are 43 and 38. What is the measure of the third angle?. 9 G. 81 H. 99 J Which method would you use to find the inverse of a conditional statement?. Negate the hypothesis only.. Negate the conclusion only.. witch the hypothesis and. Negate both the hypothesis and the conclusion. the conclusion. V Y Z W earsonx.com 157

13 4-3 riangle ongruence by and K OU K (6)() rove two triangles are congruent by applying the ide-ngle-ide, ngle-ide-ngle, ide- ide-ide, ngle-ngle-ide, and Hypotenuse-Leg congruence conditions. VOULY nalyze closely examine objects, ideas, or relationships to learn more about their nature. K (1)() nalyze mathematical relationships to connect and communicate mathematical ideas. dditional K (1)(), (1)(), (1)(G) NIL UNNING You can prove that two triangles are congruent without having to show that all corresponding parts are congruent. In this lesson, you will prove triangles congruent by using one pair of corresponding sides and two pairs of corresponding angles. ostulate 4-3 ngle-ide-ngle () ostulate ostulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. If...,, hen... heorem 4-2 ngle-ngle-ide () heorem heorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. If...,, hen... You will prove heorem 4-2 in xercise Lesson 4-3 riangle ongruence by and

14 roblem 1 K rocess tandard (1)() Using Which two triangles are congruent by? xplain. O N rom the diagram you know U V O W UV O W U V W o use, you need two pairs of congruent angles and a pair of included congruent sides. You already have pairs of congruent angles. o, identify the included side for each triangle and see whether it has a congruence marking. In UV, UV is included between U and V and has a congruence marking. In NO, O is included between and O and has a congruence marking. In W, W is included between and W but does not have a congruence marking. ince U, UV O, and V O, UV NO. roblem 2 K rocess tandard (1)() Writing a Using ecreation embers of a teen organization are building a miniature golf course at your town s youth center. he design plan calls for the first hole to have two congruent triangular bumpers. rove that the bumpers on the first hole, shown at the right, meet the conditions of the plan. an you use a plan similar to the plan in roblem 1? Yes. Use the diagram to identify the included side for the marked angles in each triangle. Given:,, and are right angles rove: : because all right angles are congruent, and you are given that. and are included sides between the two pairs of congruent angles. You are given that. hus, by. earsonx.com 159

15 roblem 3 How does information about parallel sides help? You will need another pair of congruent angles to use. hink back to what you learned in opic 3. W is a transversal here. Writing a Using Given: G K, W } K rove: W KW W K tatements easons 1) K 1) Given 2) W } K 2) Given 3) W KW 3) If lines are }, then alternate interior are. 4) W W 4) eflexive roperty of ongruence 5) W KW 5) roblem 4 etermining Whether riangles re ongruent ultiple hoice Use the diagram at the right. Which of the following statements best represents the answer and justification to the question, Is I@ UO? Yes, the triangles are congruent by. an you eliminate any of the choices? Yes. If I@ UO then and U would be corresponding angles. You can eliminate choice. No, and O are not corresponding sides. Yes, the triangles are congruent by. No, and U are not corresponding angles. he diagram shows that two pairs of angles and one pair of sides are congruent. he third pair of angles is congruent by the hird ngles heorem. o prove these triangles congruent, you need to satisfy or. and both fail because and O are not included between the same pair of congruent corresponding angles, so they are not corresponding sides. he triangles are not necessarily congruent. he correct answer is. O U I 160 Lesson 4-3 riangle ongruence by and

16 ONLIN H O W O K I and LIION XI can page for a Virtual Nerd tutorial video. etermine whether the triangles must be congruent. If so, name the postulate or theorem that justifies your answer. If not, explain. or additional support when completing your homework, go to earsonx.com U N O Z V Y W 4. Given: JG HGJ, G } JH 5. Given: Q#Q, #Q, rove: GJ HJG is the midpoint of J G 6. valuate easonableness (1)() While helping your family clean out the attic, you find the piece of paper shown at the right. he paper contains clues to locate a time capsule buried in your backyard. he maple tree is due east of the oak tree in your backyard. Will the clues always lead you to the correct spot? xplain. 7. onnect athematical Ideas (1)() nita says that you can rewrite any proof that uses the heorem as a proof that uses the ostulate. o you agree with nita? xplain. 8. Justify athematical rguments (1)(G) an you prove that the triangles at the right are congruent? Justify your answer. H rove: Q 9. Given: N, O QO 10. Given: 1 2, and rove: ON QO H bisects Q O N Q rove: H H 1 H 2 earsonx.com 161

17 11. Given: }, } rove: 12. reate epresentations to ommunicate athematical Ideas (1)() raw two noncongruent triangles that have two pairs of congruent angles and one pair of congruent sides. 13. Given } and }, name as many pairs of congruent triangles as you can. 14. reate epresentations to ommunicate athematical Ideas (1)() Use a straightedge to draw a triangle. Label it JKL. onstruct N JKL so that the triangles are congruent by. 15. rove the ngle-ngle-ide heorem (heorem 4-2). Use the diagram next to it on page In at the right, = 5, = 9, and m = 30. how that there is no congruence rule by constructing UVW with UV =, UW =, and m W = m, but with UVW X est ractice 17. uppose N and N. What additional information do you need to prove that J N by?.. J. J. 18. You plan to make a 2 ft-by-3 ft rectangular poster of class trip photos. ach photo is a 4 in.-by-6 in. rectangle. If the photos do not overlap, what is the greatest number of photos you can fit on your poster?. 4 H. 32 G. 24 J Write the converse of the true conditional statement below. hen determine whether the converse is true or false. If you are less than 18 years old, then you are too young to vote in the United tates. 162 Lesson 4-3 riangle ongruence by and

18 echnology Lab xploring and U WIH LON 4-3 K (5)(), (1)() o far, you know four ways to conclude that two triangles are congruent,,, and. It is good mathematics to wonder about the other two possibilities. 1 onstruct Use geometry software to construct > and >. onstruct to form. onstruct a line parallel to that intersects > and > at points and to form. Investigate re the three angles of congruent to the three angles of? anipulate the figure to change the positions of and. o the corresponding angles of the triangles remain congruent? re the two triangles congruent? an the two triangles be congruent? 2 onstruct onstruct >. raw a circle with center that intersects > at two points. onstruct. onstruct point on the circle and construct. Investigate ove point around the circle until is on > and forms. hen move on the circle to the other point on > to form another. ompare,, and m in the two triangles. re two sides and a nonincluded angle of one triangle congruent to two sides and a nonincluded angle of the other triangle? re the triangles congruent? If you change the measure of and the size of the circle, do you get the same results? xercises 1. ake a onjecture ased on your first investigation above, can you prove triangles congruent using? xplain. or xercises 2 4, use what you learned in your second investigation above. 2. ake a onjecture an you prove triangles congruent using? xplain. 3. anipulate the figure so that is obtuse. an the circle intersect > twice to form two triangles? Would work if the congruent angles were obtuse? xplain. 4. uppose you are given,, and. What must be true about,, and m so that you can construct exactly one? (Hint: onsider cases.) earsonx.com 163

19 4-4 Using orresponding arts of ongruent riangles K OU VOULY Ě Justify explain with logical reasoning. You can justify a K (6)() rove two triangles are congruent by applying the ide-ngle-ide, ngle-ide-ngle, ide-ide-ide, ngle-ngle-ide, and HypotenuseLeg congruence conditions. mathematical argument. Ě rgument a set of statements put forth to show the truth or falsehood of a mathematical claim K (1)(G) isplay, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. NIL UNNING If you know two triangles are congruent, then you know that every pair of their corresponding sides and angles is also congruent. roblem 1 r roving arts of riangles ongruent Given: K, K rove: K K In the diagram, which congruent pair is not marked? he third angles of both triangles are congruent. ut there is no congruence rule. o, find a congruent pair of sides. 164 / K > / Given /K > / Given > eflexive roperty of > n K > n heorem Lesson 4-4 Using orresponding arts of ongruent riangles K > orresp. parts of > are >.

20 roblem 2 K rocess tandard (1)(G) Which congruency rule can you use? You have information about two pairs of angles. Guess-andcheck and. roving riangle arts ongruent to easure istance easurement hales, a Greek philosopher, is said to have developed a method to measure the distance to a ship at sea. He made a compass by nailing two sticks together. tanding on top of a tower, he would hold one stick vertical and tilt the other until he could see the ship along the line of the tilted stick. With this compass setting, he would find a landmark L on the shore along the line of the tilted stick. How far would the ship be from the base of the tower? Given: and L are right angles, L rove: L L tatements easons 1) L 1) Given 2) 2) eflexive roperty of ongruence 3) and L are right angles. 3) Given 4) L 4) ll right angles are congruent. 5) L 5) ostulate 6) L 6) orresponding parts of s are. he distance between the ship and the base of the tower would be the same as the distance between the base of the tower and the landmark. earsonx.com 165

21 ONLIN H O W O K I and LIION XI can page for a Virtual Nerd tutorial video. or additional support when completing your homework, go to earsonx.com. 1. xplain athematical Ideas (1)(G) ell why the two triangles are congruent. Give the congruence statement. hen list all the other corresponding parts of the triangles that are congruent. 2. Given:, rove: 3. Given: O, O rove: O K L J N O 4. Justify athematical rguments (1)(G) balalaika is a stringed instrument. rove that the bases of the balalaikas are congruent. Given: NY, K JNY, K JYN rove: K JY N : It is given that two angles and the included side of one triangle are congruent to two angles and the included side of the other. o, a.? JNY by b.?. K JY because c.?. K J Y 5. Given: O, 6. Given: Y Y,, O rove: O rove: O Y nalyze athematical elationships (1)() opy and mark the figure to show the given information. xplain how you would prove j@ jq. K 7. Given: K QK, KL bisects KQ 8. Given: KL is the perpendicular bisector of Q. 9. Given: KL#Q, KL bisects KQ L Q 166 Lesson 4-4 Using orresponding arts of ongruent riangles

22 10. Justify athematical rguments (1)(G) he construction of a line perpendicular to line / through point on line / is shown. xplain why you can conclude that < > is perpendicular to /. 11. he construction of congruent to given is shown. because they are congruent radii. because both arcs have the same compass settings. xplain why you can conclude that. 12. Given: #, #,, rove: 13. Given: JK } Q, JK Q rove: KQ bisects J K J Q 14. pply athematics (1)() angoli is a colorful design pattern drawn outside houses in India, especially during festivals. Vina plans to use the pattern at the right as the base of her design. In this pattern, U, V, and Q bisect each other at O. = 6, U = 12, U V, }U, and }Q. What is the perimeter of the hexagon? In the diagram at the K K. K 15. rove: is the midpoint of K. 16. rove: K# X est ractice or xercises 17 and 18, use the diagram at the right. # and bisects j and j. 17. uppose = 17 and = 5. What is the length of? 18. uppose m = 64, and m = 16. What is m? 19. wo parallel lines q and s are cut by a transversal t. 1 and 2 are a pair of alternate interior angles and m 2 = and 3 are vertical angles. What is m 3? 20. has vertices (1, 9), (4, 3), and (x, 6). or what value of x is a right triangle with right? earsonx.com 167

23 4-5 Isosceles and quilateral riangles K OU VOULY K (5)() Use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships. K (1)() elect tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Ě ase of an isosceles triangle the non-congruent side of an isosceles triangle Ě ase angles of an isosceles triangle the two angles in an isosceles triangle that are formed by the intersection of a leg and the base Ě orollary a theorem that can be proved easily using another theorem Ě Legs of an isosceles triangle the congruent sides of an isosceles triangle Ě Vertex angle of an isosceles triangle the angle in an isosceles triangle formed by the two congruent legs Ě Number sense the understanding of what numbers mean and how they are related dditional K (5)(), (6)(), (6)() NIL UNNING he sides and angles of isosceles and equilateral triangles have special relationships. heorem 4-3 heorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Isosceles riangle heorem If... hen... or a proof of heorem 4-3, see roblem 2. heorem 4-4 heorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. onverse of the Isosceles riangle heorem If... hen... You will prove heorem 4-4 in xercise Lesson 4-5 Isosceles and quilateral riangles

24 heorem 4-5 heorem If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base. If... and hen... # and You will prove heorem 4-5 in xercise 10. orollary to heorem 4-3 orollary If a triangle is equilateral, then the triangle is equiangular. If... XY YZ ZX Y hen... X Y Z Y X Z X Z orollary to heorem 4-4 orollary If a triangle is equiangular, then the triangle is equilateral. If... X Y Z Y hen... XY YZ ZX Y X Z X Z roblem 1 K rocess tandard (1)() Using onstructions of ongruent egments onstruct congruent segments to make a conjecture about the angles opposite the congruent sides in an isosceles triangle. tep 1 onstruct an isosceles on tracing paper, with. How can folding a piece of paper help you tell if two angles are congruent? When folding the paper, congruent angles will fit exactly one on top of the other. tep 2 old the paper so that the two congruent sides fit exactly one on top of the other. rease the paper. Notice that and appear to be congruent. onjecture: ngles opposite the congruent sides in an isosceles triangle are congruent. continued on next page earsonx.com 169

25 roblem 1 continued onstruct congruent angles to make a conjecture about the sides opposite congruent angles in a triangle. tep 1 raw on tracing paper. hen construct congruent to so that intersects, resulting in a triangle. tep 2 roblem 2 old the paper so that the two congruent angles fit exactly on top of each other. Notice that the sides of the triangle opposite the congruent angles appear to be congruent. onjecture: ides opposite the congruent angles in a triangle are congruent. How are the sides and angles of an isosceles triangle related? he congruent sides of an isosceles triangle are its legs, and the third side is its base. he two congruent legs form the vertex angle, while the other two angles are the base angles. roving the Isosceles riangle heorem egin with isosceles XYZ with XY XZ. raw X, the bisector of vertex angle jyxz. Given: XY XZ, X bisects YXZ rove: Y Z tatements easons 1) XY XZ 1) Given 2) 1 2 2) efinition of angle bisector 3) X X 3) eflexive roperty of ongruence 4) XY XZ 4) ostulate 5) Y Z 5) orresponding parts of s are. X 1 2 Y Z roblem 3 What are you looking for in the diagram? o use the Isosceles riangle theorems, you need a pair of congruent angles or a pair of congruent sides. Using the Isosceles riangle heorem and its onverse Is congruent to? xplain. Yes. ince, by the onverse of the Isosceles riangle heorem. Is j congruent to j? xplain. Yes. ince, by the Isosceles riangle heorem. 170 Lesson 4-5 Isosceles and quilateral riangles

26 What does the diagram tell you? ince, is isosceles. ince, bisects the vertex angle of the isosceles triangle. roblem 4 Using lgebra What is the value of x? ince, by the Isosceles riangle heorem,. o m = 54. ince bisects, you know by heorem 4-5 that #. o m = 90. m + m + m = x = 180 ubstitute. x = 36 riangle ngle-um heorem ubtract 144 from each side x roblem 5 inding ngle easures esign What are the measures of j, j, and j in the photo image at the right? K rocess tandard (1)(G) he triangles are equilateral, so they are also equiangular. ind the measure of each angle of an equilateral triangle. and are both angles in an equilateral triangle. Let a = measure of one angle. 3a = 180 a = 60 m = m = 60 Use the ngle ddition ostulate to find the measure of. oth and are angles in an equilateral triangle. o m = 60 and m = 60. ubstitute into the above equation and simplify. m = m + m m = m = 120 earsonx.com 171

27 ONLIN H O W O K I and LIION XI can page for a Virtual Nerd tutorial video. omplete each statement. xplain why it is true. 1. V? V or additional support when completing your homework, go to earsonx.com. 2. U? YX 3. VU? U Y 4. VYU? W X 5. Justify athematical rguments (1)(G) builder using the truss shown at the right claims that will have the same measure as. and represent identical beams, and bisects. Is the builder correct? Justify your answer. onnect athematical Ideas (1)() ind the values of x and y x y x 52 y x y 9. pply athematics (1)() ach face of the Great yramid at Giza is an isosceles triangle with a 76 vertex angle. What are the measures of the base angles? 10. rove heorem 4-5. Use the diagram next to it on page 169. Given isosceles JKL with base JL, find each value. 11. If m L = 58, then m LKJ =?. 12. If JL = 5, then L =?. 13. If m JK = 48, then m J =?. 14. If m J = 55, then m JK =?. 15. nalyze athematical elationships (1)() triangle has angle measures x + 15, 3x - 35, and 4x. What type of triangle is it? e as specific as possible. Justify your answer. 16. upply the missing information in this statement of the onverse of the Isosceles riangle heorem. hen write a proof. egin with Q with Q. raw a.?, the bisector of Q. Given: Q, b.? bisects Q rove: Q K J L Q 172 Lesson 4-5 Isosceles and quilateral riangles

28 17. a. pply athematics (1)() In the diagram at the right, what type of triangle is formed by the cables of the same height and the ground? b. What are the two different base lengths of the triangles? c. How is the tower related to each of the triangles? 18. nalyze athematical elationships (1)() he length of the base of an isosceles triangle is x. he length of a leg is 2x - 5. he perimeter of the triangle is 20. ind x. or each pair of points, there are six points that could be the third vertex of an isosceles right triangle. ind the coordinates of each point ft 800 ft 600 ft 400 ft 200 ft 19. (4, 0) and (0, 4) 20. (0, 0) and (5, 5) 21. (2, 3) and (5, 6) 0 ft adio tower 1009 ft tall ables 450 ft 550 ft 22. reate epresentations to ommunicate athematical Ideas (1)() Use at the right. a. onstruct a right triangle with one leg congruent to and hypotenuse congruent to. b. onstruct a right triangle with one leg congruent to and hypotenuse congruent to. c. raw a new right triangle with different side lengths than. epeat parts (a) and (b) for your new right triangle. d. Use your results to make a conjecture about congruence criteria for right triangles. X est ractice 23. In isosceles, the vertex angle is. What can you prove?. =. m = m What is the exact area of the base of a circular swimming pool with diameter 16 ft? ft 2 G ft 2 H. 64p ft 2 J. 256p ft uppose and are nonright triangles. If and, what else do you need to know to prove? xplain. earsonx.com 173

29 K OU 4-6 ongruence in ight riangles VOULY K (6)() rove two triangles are congruent by applying the ide-ngle-ide, ngle-ide-ngle, ide-ide-ide, ngle-ngle-ide, and Hypotenuse- Leg congruence conditions. K (1)() nalyze mathematical relationships to connect and communicate mathematical ideas. Hypotenuse the side opposite the right angle Legs of a right triangle the two sides other than the hypotenuse in a right triangle nalyze closely examine objects, ideas, or relationships to learn more about their nature NIL UNNING You can prove that two triangles are congruent without having to show that all corresponding sides and angles are congruent. In this lesson, you will prove right triangles congruent by using one pair of right angles, a pair of hypotenuses, and a pair of legs. heorem 4-6 Hypotenuse-Leg (HL) heorem heorem If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. If... Q and XYZ are right s, XZ, and Q XY Q X Y Z hen... Q XYZ or a proof of heorem 4-6, see the eference section on page 683. Key oncept onditions for HL heorem o use the HL heorem, the triangles must meet three conditions. onditions here are two right triangles. he triangles have congruent hypotenuses. here is one pair of congruent legs. 174 Lesson 4-6 ongruence in ight riangles

30 roblem 1 r K rocess tandard (1)() Using the HL heorem On the basketball backboard brackets shown below, and are right angles and. re and congruent? xplain. How can you visualize the two right triangles? Imagine cutting along. On either side of the cut, you get triangles with the same leg. r :V HJWO UIU O JHIU OH 4 ű űoű űűjhiuűujoh r 5I IZQUOV G UI UX JHIU UJOH O :VűűHJWOűUIUű r J NNO H G UI O Z UI 3GYJW 1QUZ G $OHVO : Z UI )- 5IN earsonx.com 175

31 roblem 2 Writing a Using the HL heorem How can you get started? Identify the hypotenuse of each right triangle. rove that the hypotenuses are congruent. Given: bisects at, #, #, rove: bisects. Given Given and are right. ef. of lines ef. of bisector and are right. ef. of right triangle HL heorem Given ONLIN H O W O K I and LIION XI can page for a Virtual Nerd tutorial video. 1. Justify athematical rguments (1)(G) opy the flow chart and complete the proof. or additional support when completing your homework, go to earsonx.com. Given:, rove: and are. Given and are right. a. and are right. b. and are supplementary. that form a linear pair are supplementary. c. d. e. 2. tudy xercise 1. an you prove that without using the HL heorem? xplain. 3. xplain athematical Ideas (1)(G) omplete the paragraph proof. Given: and are right angles, rove: : It is given that and are right angles. o, a.? by the definition of right triangles. b.?, because of the eflexive roperty of ongruence. It is also given that c.?. o, by d.?. 176 Lesson 4-6 ongruence in ight riangles

32 4. Given: HV#G, GH V, I is the midpoint of HV rove: IGH IV H G I 5. Given: J, #J, #J, is the midpoint of J rove: J onnect athematical Ideas (1)() or what values of x and y are the triangles congruent by HL? x x 3 3y y 1 V 3y x y x J y 5 x 5 8. pply athematics (1)() and Q are right triangular sections of a fire escape, as shown. Is each story of the building the same height? xplain. 9. onnect athematical Ideas (1)() ha! exclaims your classmate. here must be an H heorem, sort of like the HL heorem! Is your classmate correct? xplain. 10. Given: LN is isosceles with base N, N#NL, Q#L, L QL rove: NL QL L N Q Q reate epresentations to ommunicate athematical Ideas (1)() opy the triangle and construct a triangle congruent to it using the given method HL earsonx.com 177

33 15. Given: GK is isosceles with base G, L and are right angles, and K is the midpoint of L. rove: LG L K 16. Given: LO bisects LN, O#L, ON#LN rove: LO LNO O L N G 17. Justify athematical rguments (1)(G) re the triangles at the right congruent? xplain nalyze athematical elationships (1)() or xercises 18 and 19, use the figure at the right. 18. Given: #, #, is equilateral rove: 19. Given:, #, # an you prove that is equilateral? xplain. X est ractice 20. You often walk your dog around the neighborhood. ased on the diagram at the right, which one of the following statements about distances is true?. H = LH. H 7 LH. H = H. H 6 H chool () ark () afé () Library (L) Home (H) 21. In equilateral XYZ, name four pairs of congruent right triangles. xplain why they are congruent. X Q Y Z 178 Lesson 4-6 ongruence in ight riangles

34 4-7 ongruence in Overlapping riangles K OU VOULY Ě epresentation a way to display or describe information. K (6)() rove two triangles are congruent by applying the ide-ngle-ide, ngle-ide-ngle, ideide-ide, ngle-ngle-ide, and Hypotenuse-Leg congruence conditions. You can use a representation to present mathematical ideas and data. K (1)() reate and use representations to organize, record, and communicate mathematical ideas. dditional K (1)(), (1)(G) NIL UNNING You can sometimes use the congruent corresponding parts of one pair of congruent triangles to prove another pair of triangles congruent. his often involves overlapping triangles. roblem 1 r Identifying ommon arts What common angle do and share? How can you see an individual triangle in order to redraw it? Use your finger to trace along the lines connecting the three vertices. hen cover up any untraced lines. e eparate and redraw and. he common angle is. h earsonx.com 179

35 roblem 2 K rocess tandard (1)() Using ommon arts Given: ZXW YWX, ZWX YXW Z Y rove: ZW YX W X ZXW YWX and ZWX YXW ZWX and YXW are overlapping triangles. Z Y how ZWX YXW. hen use corresponding parts of congruent triangles to prove ZW YX. W X W X ZXW YWX Given roblem 3 WX WX eflexive rop. of ZWX YXW Given ZWX YXW ZW YX orresp. parts of are. Using wo airs of riangles How do you choose another pair of triangles to help in your proof? Look for triangles that share parts with G and J and that you can prove congruent. In this case, first prove. Given: In the origami design, is the midpoint of and. rove: G J : is the midpoint of and, so and. because vertical angles are congruent. herefore, by. because corresponding parts of congruent triangles are congruent. G J because vertical angles are congruent. herefore, G J by. G J 180 Lesson 4-7 ongruence in Overlapping riangles

36 roblem 4 K rocess tandard (1)(G) eparating Overlapping riangles Given:, Which triangles are useful here? If X X, then X X because they are corresponding parts. If, you will have enough information to show X X. rove: X X X X tatements easons 1) 1) Given 2) 2) Given 3) 3) ase of an isosceles are. 4) 4) eflexive roperty of 5) 5) 6) 6) orresp. parts of s are. 7) X X 7) Vertical angles are. 8) X X 8) 9) X X 9) orresp. parts of s are. ONLIN H O W O K I and LIION XI can page for a Virtual Nerd tutorial video. In each diagram, the red and blue triangles are congruent. Identify their common side or angle. or additional support when completing your homework, go to earsonx.com. 1. K X L N G Y W Z eparate and redraw the indicated triangles. Identify any common sides or angles. 4. Q and Q 5. and 6. JKL and LK Q K L O J earsonx.com 181

37 7. Justify athematical rguments (1)(G) omplete the flow proof. Given:, Q V rove: Q V V Q a. Q V b. Q V d. Q V e. Q V c. 8. Given: U, U rove: U 9. Given: Q U, Q U rove: Q U Q U U W V 10. Given: 1 2, 3 4 rove: Q QU 11. Given:, is the midpoint of rove: G G Q U 12. xplain athematical Ideas (1)(G) In the diagram at the right, V, VU, and QV. Which two triangles are congruent by? xplain. W X Q V U 13. Identify a pair of overlapping congruent triangles in the diagram. hen use the given information to write a proof to show that the triangles are congruent. Given:, 182 Lesson 4-7 ongruence in Overlapping riangles

38 14. pply athematics (1)() he figure at the right is part of a clothing design pattern, and it has the following relationships. G# # }}G m = 50 is isosceles with base. a. ind the measures of all the numbered angles in the figure. b. uppose. Name two congruent triangles and explain how you can prove them congruent. 15. Given:, rove: 16. Given: Q#, Q bisects, Q bisects VQ rove: VQ Q Q 17. reate epresentations to ommunicate athematical Ideas (1)() raw a quadrilateral with }, }, and diagonals and intersecting at. Label your diagram to indicate the parallel sides. a. List all the pairs of congruent segments in your diagram. b. xplain how you know that the segments you listed are congruent. 4 H 8 J 9 I V 7 G X est ractice 18. ccording to the diagram at the right, which statement is true?. H GH by. G by. H GH by. G by 19. is isosceles with base. If m = 37, what is m?. 37 G. 74 H. 106 J. 143 G H 20. Which word correctly completes the statement ll? angles are congruent?. adjacent. supplementary. right. corresponding 21. In the figure, LJ}GK and is the midpoint of LG. a. opy the diagram. hen mark your diagram with the given information. b. rove LJ GK. c. an you prove that LJ GK another way? xplain. L J K G earsonx.com 183

39 opic 4 eview OI VOULY p. 168 p. 168 p 148 p. 168 p. 174 p 174 p. 168 p. 168 heck Your Understanding hoose the correct term to complete each sentence. 1. he two congruent sides of an isosceles triangle are the?. 2. he side opposite the right angle of a right triangle is the?. 3.? to a theorem is a statement that follows immediately from the theorem. 4.? have congruent corresponding parts. 4-1 ongruent igures Quick eview ongruent polygons have congruent corresponding sides and angles. When you name congruent polygons, always list corresponding vertices in the same order. xample HIJK@Q. Write all possible congruence statements. he order of the parts in the congruence statement tells you which parts correspond. ides: HI Q, IJ Q, JK, KH ngles: H, I Q, J, K xercises UV@KLNO. omplete the congruence statements. 5.? 6. N? 7. L? 8. VU? WXYZ Q. ind each measure or length. W X Y 8.6 Z 9. m 10. Q 11. WX Q m Z 13. m X 14. m 184 opic 4

40 4-2 and 4-3 riangle ongruence by,,, and Quick eview You can prove triangles congruent with limited information about their congruent sides and angles. ostulate or heorem ide-ide-ide () ide-ngle-ide () ngle-ide-ngle () ngle-ngle-ide () You need three sides two sides and an included angle two angles and an included side two angles and a nonincluded side xercises 15. In H, what angle is included between H and? 16. In O, what side is included between and? Which postulate or theorem, if any, could you use to prove the two triangles congruent? If there is not enough information to prove the triangles congruent, write not enough information xample What postulate would you use to prove the triangles congruent? You know that three sides are congruent. Use. 4-4 Using orresponding arts of ongruent riangles Quick eview Once you know that triangles are congruent, you can make conclusions about corresponding sides and angles because, by definition, corresponding parts of congruent triangles are congruent. You can use congruent triangles in the proofs of many theorems. xample How can you use congruent triangles to prove jq@j? Q W ince QW VK by, you know that Q because corresponding parts of congruent triangles are congruent. K V xercises How can you use congruent triangles to prove the statement true? 21. V YW 22. V Y W KN L X K N L earsonx.com 185

41 4-5 Isosceles and quilateral riangles Quick eview If two sides of a triangle are congruent, then the angles opposite those sides are also congruent by the Isosceles riangle heorem. If two angles of a triangle are congruent, then the sides opposite those angles are congruent by the onverse of the Isosceles riangle heorem. quilateral triangles are also equiangular. xample What is mjg? ince G, G by the Isosceles riangle heorem. o m G = 30. G 30 xercises lgebra ind the values of x and y x 4 y y x 25 y 7 x 125 x y 4-6 ongruence in ight riangles Quick eview If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent by the Hypotenuse-Leg (HL) heorem. xercises Write a proof for each of the following. 29. Given: LN#K, KL L rove: KLN LN L xample Which two triangles are congruent? xplain. N L Z ince and XYZ are right triangles with congruent legs, and YZ, XYZ by HL. X Y K 30. Given: #Q, Q#Q, Q N rove: Q Q Q 186 opic 4 eview

42 4-7 ongruence in Overlapping riangles Quick eview o prove overlapping triangles congruent, you look for the common or shared sides and angles. xample eparate and redraw the overlapping triangles. Label the vertices. xercises Name a pair of overlapping congruent triangles in each diagram. tate whether the triangles are congruent by,,,, or HL Q I G H earsonx.com 187

43 opic 4 K umulative ractice ultiple hoice ead each question. hen write the letter of the correct answer on your paper. 1. Given: }, rove: : ince }, and by the orresponding ngles heorem. ince, by the ransitive roperty of ongruence. Which theorem or definition proves that?. Isosceles riangle heorem. onverse of Isosceles riangle heorem. lternate Interior ngles heorem. efinition of congruent segments 2. Which statement must be true for two polygons to be congruent?. ll the corresponding sides should be congruent. G. ll the corresponding sides and angles should be congruent. H. ll the corresponding angles should be congruent. J. ll sides in each polygon should be congruent. 3. If, which of the following must be true? Given: 1 2, What additional information do you need to prove by?. H. 5 6 G. J. 5. Which of the following statements is true?. oint, line, and plane are undefined terms.. theorem is an accepted statement of fact.. Vertical angles are congruent is a definition.. postulate is a conjecture that is proven. 6. Which condition allows you to prove that / } m? H. 3 4 G. 2 8 J he measure of one base angle of an isosceles triangle is 23. What is the measure of the vertex angle? m 188 opic 4 K umulative ractice

44 Gridded esponse 8. What is the value of x in the figure below? What is the value of x in the figure below? WXYZ. What is WX? (2x 5) onstructed esponse (5x 6) Y 4 X W 11. Is GK HK? xplain. G H K L J 12. Write a proof for the following. Given:, rove: 2 Z 13. Write a proof for the following. Given: G, } G rove: G G 14. Write a proof for the following. Given: LN bisects OL and ON. rove: ON N 15. Isosceles, with right, has a point on such that #. What is the relationship between and? xplain. 16. ead this excerpt from an online news article. Halley s omet can be seen periodically at its perihelion, the shortest distance from the sun during its orbit. ark wain was born two weeks after the comet s perihelion. In his biography he said, I came in with Halley s omet in It is coming again next year, and I expect to go out with it. wain died in 1910, the day after the comet s perihelion. he most recent sighting of Halley s omet was in Its next appearance is expected in a. ake a conjecture about the year in which Halley s omet will appear after xplain your reasoning. b. How confident are you about your conjecture? xplain. 17. he coordinates of the vertices of rectangle LNK are L(-2, 5), (2, 5), N(2, 3), and K(-2, 3). he coordinates of the vertices of rectangle Q are (3, 0), Q(3, -3), (1, -3), and (1, 0). re these two rectangles congruent? xplain why or why not. If not, how could you change the vertices of one of the rectangles to make them congruent? O G L N earsonx.com 189